The length constant can be related to the electrical properties of the axon according to cable theory because nerve fibers have many of the properties of an electrical cable. In a perfect cable, the insulation surrounding the core conductor prevents all loss of current to the surrounding medium, so that a signal is transmitted along the cable with undiminished strength. If an unmyelinated nerve fiber (discussed later) is compared to an electrical cable, the plasma membrane equates to the insulation and the cytoplasm as the core conductor, but the plasma membrane is not a perfect insulator. Thus the spread of signals depends on the ratio of the membrane resistance to the axial resistance of the axonal cytoplasm (ra). When the ratio of rm to ra is high, less current is lost across the plasma membrane per unit of axonal length, the axon can function better as a cable, and the distance that a signal can be conveyed electrotonically without significant decrement is longer. A useful analogy is to think of the axon as a garden hose with holes poked in it. The more holes there are in the hose, the more water leaks out along its length (analogous to more loss of current when rm is low) and the less water is delivered to its nozzle. According to cable theory, the length constant can be related to axonal resistance and is equal to rm/ra . This relationship can be used to determine how changes in axonal diameter affect the length constant and, hence, how the decay of electrotonic potentials varies. An increase in the diameter of the axon reduces both ra and rm. However, rm is inversely proportional to diameter (because it is related to the circumference of the axon), whereas ra varies inversely to the diameter squared (because it is related to the cross-sectional area of the axon). Thus ra decreases more rapidly than rm does as axonal diameter increases, and the length constant therefore increases (
Fig. 5.3 ˜ = Distance over which response decays to 1 or ~ 37% of original size (Vo)
Vo/e Vo/e Vo/e Vo/e Large-diameter axon ra is lowra is high˜Vo ˜˜˜˜ = 3 mm ˜ = 1 mm 1. 2.
•Fig. 5.3 Comparison of the Length Constant to Axon Diameter. Notethattheincreaseindiameterisassociatedwithadecreaseinaxialresistanceoftheaxonalcytoplasm(ra)andanincreaseinthelengthconstant(λ).(RedrawnfromBlankenshipJ.Neurophysiology. Philadelphia:Mosby;2002.)
CHAPTER 5 Generation and Conduction of Action Potentials
In sum, in the passive domain, the membrane response to electrical stimuli is essentially identical to that of a circuit composed of passive electrical elements, and it can thus be characterized by the length and time constants of the membrane, which will determine how far and how rapidly electrical signals at one point in the cell spread to other parts.
The Local Response
With regard to the experiment shown in
Fig. 5.1, if larger depolarizing current pulses are injected, the voltage response of the membrane no longer resembles that of a passive RC circuit. This is most easily observed with pulses that elicit depolarizations either just below or to the threshold membrane potential for an action potential but fail to evoke an action potential (tracings 0.89 and 1.0; the threshold membrane potential can be defined as the voltage at which the probability of evoking an action potential is 50%). In these cases, the voltage response shape is altered from that of the passive responses because the stimulus has changed the membrane potential sufficiently to cause the opening of significant numbers of voltage-sensitive Na+ channels (described later).
Also, note the upward deviation from linearity for the corresponding points in the I-V curve (see